SELF-BALANCING BOT USING CONCEPT
OF INVERTED PENDULUM
Pratyusa kumar Tripathy (109EC0427)
Department of Electronics and Communication Engineering
National Institute of Technology Rourkela
Rourkela- 769008,India
SELF-BALANCING BOT USING
CONCEPT OF INVERTED PENDULUM
A project submitted in partial fulfilment of the requirements for the degree of
Bachelor of Technology
in
Electronics and Communication Engineering
by
Pratyusa kumar Tripathy
(Roll 109EC0427)
under the supervision of
Prof. S. K. Das
Department of Electronics and Communication Engineering
National Institute of Technology Rourkela
Rourkela -769008, India
Electronics and Communication Engineering
National Institute of Technology Rourkela Rourkela-769008, India.www.nitrkl.ac.in
Prof S. K. Das May 13th, 2013
Certificate
This is to certify that the work in the Project entitled self-balancing robot using
concept of inverted pendulum by Pratyusa kumar Triparthy, is a record of
an original research work carried out by him under my supervision and guidance
in partial fulfilment of the requirements for the award of the degree of Bachelor
of Technology in Electronics and Communication Engineering. Neither this thesis
nor any part of it has been submitted for any degree or academic award elsewhere.
Prof S. K. Das
Acknowledgment
I would like to express my sincere gratitude and thanks to my supervisor Prof. S. K. Das for
his constant guidance, encouragement and extreme support throughout the course of this
project. I am thankful to Electronics and Communication Engineering Department to provide us
with the unparalleled facilities throughout the project. I am thankful to Ph.D Scholars for their
help and guidance in completion of the project. I am thankful to our batch mates and friends
specifically Debabrata Mahapatra and my super senior Subhranshu Mishra for their support and
being such a good company. I extend my gratitude to researchers and scholars whose papers
and thesis have been utilized in our project. Finally, I dedicate my thesis to our parents for their
love, support and encouragement without which this would not have been possible.
Pratyusa Kumar Tripathy
Abstract
Self-balancing robot is based on the principle of Inverted pendulum, which is a
two wheel vehicle balances itself up in the vertical position with reference to the ground. It
consist both hardware and software implementation. Mechanical model based on the state
space design of the cart, pendulum system. To find its stable inverted position, I used a generic
feedback controller (i.e. PID controller). According to the situation we have to control both
angel of pendulum and position of cart. Mechanical design consist of two dc gear motor with
encoder, one arduino microcontroller, IMU (inertial mass unit) sensor and motor driver as a
basic need. IMU sensor which consists of accelerometer and gyroscope gives the reference
acceleration and angle with respect to ground (vertical), When encoder which is attached with
the motor gives the speed of the motor. These parameters are taken as the system parameter
and determine the external force needed to balance the robot up.
It will be prevented from falling by giving acceleration to the wheels according to its
inclination from the vertical. If the bot gets tilts by an angle, than in the frame of the wheels;
the centre of mass of the bot will experience a pseudo force which will apply a torque opposite
to the direction of tilt.
CONTENT
Certificate i
Acknowledgement ii
Abstract iii
List of figures vi
List of table vii
Chapter 1 1
Introduction 2
Chapter 2 4
ANALYSIS OF SYSTEM DYNAMICS AND DESIGN OF AN APPROXIMATE MODEL 5
Chapter 3 15
PID controller and optimisation 16
Chapter 4 21
Simulation 22
Chapter 5 27
kalman filter(the estimator and predictor) 28 Chapter 6 34
Hardware implementation 35
Chapter 7 46
Real time implementation 47
Chapter 8 50
Result and graph 51
Chapter 9 60
Conclusion 61
Bibliography 62
List of figures
FIG 2.1 A model of an cart-pendulum system
Fig 2.2 force analysis of the system
Fig 3.1 PID control parameters
Fig 3.2 system with PID controller
Fig 3.3 the cart pendulum system
Fig 4.1 simulation of open loop
Fig 4.2simulation of closed loop transfer function
Fig 5.1 EKF ALL STEP AS A GRAPH
Fig 6.1 PIN diagram of arduino mega 2560
Fig 6.2 IMU sensor angle
Fig 6.3: Euler angle transformtation.
Fig 6.4 High torque motor
Fig 8.1 open loop impulse response and open loop step response
Fig 8.2 pole location and root locus
Fig 8.3 Control using LQR model
Fig 8.4 Control using LQR model with tuning parameters
Fig 8.5 control in addition of pre-compensation
Fig 8.6 cart pendulum system in unstable condition
Fig 8.7 self-balanced Pendulum cart system
Fig 8.8 output of IMU sensor
Fig 8.9 output of shaft encoder
List of Tables
Table 2.1 cart pendulum system parameters
Table 3.1 PID parameter effect comparison
Table 6.1 Arduino mega 2560 configuration
Table 6.2 motor specification
Table 6.3 interfacing of motor
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Chapter 1
Introduction
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Chapter 1
Introduction
To make a self-balancing robot, it is essential to solve the inverted pendulum problem or
an inverted pendulum on cart. While the calculation and expressions are very complex, the goal
is quite simple: the goal of the project is to adjust the wheels’ position so that the inclination
angle remains stable within a pre-determined value (e.g. the angle when the robot is not outside
the premeasured angel boundary). When the robot starts to fall in one direction, the wheels
should move in the inclined direction with a speed proportional to angle and acceleration of
falling to correct the inclination angle. So I get an idea that when the deviation from equilibrium
is small, we should move “gently” and when the deviation is large we should move more
quickly.
To simplify things a little bit, I take a simple assumption; the robot’s movement should
be confined on one axis (e.g. only move forward and backward) and thus both wheels will move
at the same speed in the same direction. Under this assumption the mathematics become much
simpler as we only need to worry about sensor readings on a single plane. If we want to allow the
robot to move sidewise, then you will have to control each wheel independently. The general
idea remains the same with a less complexity since the falling direction of the robot is still
restricted to a single axis.
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Basic Aim:
To demonstrate the methods and techniques involved in balancing an unstable robotic
platform on two wheels.
To design a complete digital control system with the state space model that will provide
the needed.
To complete the basic signal processing which will be required in making a unicycle.
A concept to start with:
Ultimate aim to start this project is the unicycle. The basics of the project basically lie on
the inclination angle. Also at the very first I thought of making a self-balancing platform, which
used data from accelerometer and gyroscope. A self-balancing bot is an advanced version of
this platform. Self-balancing bot includes the basic signal processing part (which uses kalman
filter and compensatory filter) of the unicycle.
Objective:
ANALYSIS OF SYSTEM DYNAMICS AND DESIGN OF AN APPROXIMATE MODEL
System synthesis
Simulation
Optimization
Hardware implementation
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Chapter 2
ANALYSIS OF SYSTEM DYNAMICS AND
DESIGN OF AN APPROXIMATE MODEL
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Chapter 2
2.1 Requirement Analysis
The system in my project consists of an inverted pendulum mounted to a cart which is
stable by its own wheel. The inverted pendulum system is already an exclusive example that u
can commonly find it in control system reference and research literature. Its idea concludes in
part from the fact that it is not controllable for all values of system parameter, that is, the
pendulum will simply can’t control itself on upright position. Its dynamics of the system
equation are nonlinear. The main fundamentals of the control system are to balance the
inverted pendulum by applying a force to the hinge point. A real-world example that relates
directly to this inverted pendulum system is the Segway vehicle.
In this case we will consider a two-dimensional problem where the pendulum is
constrained to move in the vertical plane shown in the figure 2.1. For this system, the control
input was the force that moves the cart horizontally and the outputs are the angular position
of the pendulum and the it at a distance x from the origin..
By taking this example we have to take a system that contains all the experimentals
components with all there charcterstics. After analysing the system we will get the eqautions
which helps to derive the state space model.
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FIG 2.1 A model of an cart-pendulum system
Component Abbreviation Value
mass of cart M 1 K.G.
Mass of pendulum m 1 K.G.
coefficient of friction on
wheel b 0.1 N/M/sec
length to pendulum from the
hinge point l 0.3 M
pendulum angle from
vertical (down) Φ To be calculated
moment of inertia of the
pendulum I 0.025 K.G.*m^2
force applied to the cart F Will be given
Table 2.1: cart pendulum system parameters
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For the PID, frequency response and root locus sections of this problem, we are
interested in the control of the pendulum angel from vertical and pendulum’s position. Using
transfer function which is best-suited for single-input, single-output (SISO) systems you can’t
control both the system output. Therefore, the design criteria deal with the cart's position and
pendulum angle simultaneously. We can, however, assume the controller's effect on the cart's
position after the controller has been designed by hit and trial method. In the next sections, we
will design a controller to keep the pendulum to a vertically upward position when it will
undergo a sudden force. Specifically, the design criteria are that the pendulum restores its upright
position within 2 seconds and that the pendulum never inclined more than 0.08 radians away
from vertical(that is controllable) after being disturbed by a force of magnitude 2 Nsec. The
pendulum will initially begin from any controllable position
But employing state-space design techniques, we are more confined and eager to choose
a multi-output system. In our case, the inverted_ pendulum system on a cart is single-input (only
external), multi-output (SIMO). Therefore, for the state-space model of the Inverted Pendulum
on a cart, we will control both the cart's position and pendulum's angle. To make the design more
accurate and well defined in this section, we will check the parameters a 0.2-meter step in the
cart's desired position. Under these conditions, it is expected that the cart achieve its stable
position within 5 seconds and have a rise time under 0.2 seconds. It is also desired that the
pendulum archive to its vertical position in under 2 seconds, and further, that the pendulum angle
not travel more than 30 degrees (0.35 radians) way from the vertically upward.
In summary, the design requirements for the inverted pendulum state-space example are:
Rise time for of less than 0.5 seconds
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Settling time for and of system equation less than 2 seconds
Steady-state error of less than 2% for and
Pendulum angle never more than 20 degrees (0.35 radians) from the vertical
2.2 Force Analysis and Equation
Figure 2.2:force analysis
By taking the total force along the horizontal axis of the cart, we got the equation:
F = H +b + M 2.1 Equation of Motion:
2.2
Angular acceleration:
2.3
22 2
2sin cos( )sin
dml mgl mg A t
dt
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Total angular force must be ‘0’:
2.4
Note: summing the forces in the vertical direction for the cart, but we can’t get any useful
information.
By taking the Sum of the forces in the free-body diagram of the pendulum cart problem in the
horizontal direction, we got the following expression for the reaction force .
2.5
By substituting equation 2.5 in the equation 2.1, we will get one of the two governing equations
of motion for the system.
2.6
2 2
2cos( ) sin 0
d g At
dt l l
2
2
0
0
( ) sin
d g
dt l
t t
Pivot
term 2
0
g
l
2
2
0
0
( ) exp( )
d g
dt l
t t
29.81m.s
19cm
g
l
1
0 7.19 rad.s
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To find the second equation of motion for this system, we have to take take the sum of the
forces perpendicular to the pendulum original position. By solving the equation along
pendulum axis greatly simplifies the mathematics. Then the equation is:
2.7
To solve for the and terms in the equation2.5 and 2.6, sum the angular moments about the
centroid of the pendulum. The equation is found out as:
2.8
Combining equation 2.7 and 2.8, we get the second governing equation.
2.9
The analysis and control design techniques only can be applied in the above problem statement
apply only to linear systems. So to solve these equations this set of equations needs to be
linearized. Simply, we will assume that the system stays within a small neighborhood of this
equilibrium; therefore, we will linearize the equations about the vertically upward equilibrium
position, = . This assumption is one type of reasonably valid since under control we have to
monitor that the pendulum should not deviate more than 20 degrees from the vertically upward
position (uncontrollable). Let defined as the deviation of the pendulum from vertically upward
position, that is, = + . Again assuming a small deviation ( ) from vertical equilibrium, we can
use the following small angle approximations of the nonlinear functions in our system equations:
2.10
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2.11
2.12
By applying the above equation 2.11 and 2.12 into nonlinear equations 2.6 and 2.9, we come
to the end with two linearized equations of motion. Note has been substituted for the input
.
2.13
2.14
Final equation of motion and torque
• F = H +b + M • V = mg + ml cos + mlsin .
• H = m + mlsin mlcos .
• I. = mglcos
2.3 Transfer function
To find the transfer functions of the system equations, we have to first take the Laplace
transform of the system equations assuming zero initial conditions. The Laplace’s equations used
are,
The Laplace transform is defined by
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Then the equation becomes,
2.15
2.16
That a transfer function of a system represents the relationship between a single input and a
single output for every equation. To find the first transfer function for the pendulum angel and
an input of U(s) we have to eliminate X(s) from the equations 2.15 and 2.16.Solving the
equation for X(s).
2.17
Then substitute the equation 2.17 into the second equation.
2.18
Manipulating, the transfer function is then the following
2.19
Where,
2.20
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From the transfer function shown in the equation 2.18 and 2.19 it can be seen that there is both a
pole and a zero at the origin. These can be canceled and the transfer function becomes the
following.
2.21
Second, the transfer function with the cart’s position will be derived in a same way to reach at
the following.
2.22
2.4 State-space model
The transfer function represented as equations of motion derivation of commanding equations
can also be represented in state-space form if they can arranged into a series of first order and
second order differential equations. They can then be put into the standard matrix, because the
equations are linear.
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The control matrix has 2 columns because both the pendulum's position and the cart's position
are part of the system output. Basically, the cart's position is the first necessary of the output
and the pendulum's angle from vertically upward position is the second element of Y.
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Chapter 3
PID CONTROLLER AND OPTIMISATION
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Chapter 3
3.1 PID controller Overview
A proportional-integral-derivative controller is a generic feedback controller. PID controller processes
the "error" as the difference between a measured output and a desired given references and tries to
minimize the error by adjusting the control parameters…
Figure 3.1:PID control parameters
In the time-domain analysis, the output of a PID controller, which is proportional to control
input is given by:
3.1
Giving look towards how the PID controller works in a closed-loop system using the system
variable. ‘e’ represents the system error due to both system noise and measurement noise, the
difference between the desired output value and the actual output produced. This error signal
is given to the PID controller, and the controller determines both the derivative and the integral
of this error. The input to the plant should be the summation of derivative constant multiplied
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by derivative error, proportional constant times the proportional error and integral times the
integral error.
When control signal ( ) will sent to the plant as the only input, and the output ( ) is obtained
according to the given input.. The new output ( ) is then given back and subtracted to the
desired reference to find error signal ( ) again and the loop continues. The PID takes this error
and calculates its control constants.
The transfer function of a PID controller is found by taking the Laplace transform of Eq.(1).
3.2
Where
= Proportional gain
= Integral gain
= Derivative gain
The structure of a plant:
Fig 3.2: system with PID controller
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3.2 Effects of control parameters on the close loop system
Due to proportional controller, we will have reduced the rise time but no effect on
steady state error. An integral control ( ) reduces the steady-state error for step input, but
negative effect on rise time. A derivative increases the stability of the system as well as reduces
the overshoot.
Closed loop responce
Rise time Over shoot Settling time Steady state error
Kp Decrease Increase Small Change Decrease
Ki Decrease Increase Increase eliminate
Kd Small Change Decrease Decrease No change
Table 3.1:PID parameter effect comparision.
3.3 PID controller design with state space
The easiest method one should attempt to make the pendulum balanced is to rotate the
wheels in the inclined direction until the inclination angle approaches to zero where the
pendulum is in balance. Basically, the rotation speed of the wheel must be proportional to
the inclination angle (e.g. move faster when the inclination is more and vice versa) so that
the robot move with a greater settle time. This is called the simplest PID control with
neglecting both the I and the D terms.
Proceeding to the next step in the design process, we have to find state-feedback
control gains represented in a vector assuming that we are well aware (i.e. can
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measure) all the state variables (four state variables are there). There is various
methods to do it. If you know the desired closed-loop pole locations, we can use the
advanced control theory. We can also use the “lqr” command which returns the optimal
controller gain by heat and trial method for a linear plant, cost function must be power
of 2 at most and initial conditions must equal to zero .
We have to check that the system is controllable before design a controller. By meeting
all of this property of controllability means that we can set the state of the system
anywhere in the controllable region (under the physical constraints of the system). The
system to meet all the conditions to be completely state controllable, the rank of the
controllability is the number of independent rows (or columns).
3.3
The controllability matrix of the system is shown by equation 3.3. The number of power
indicates to the number of state variables of the system. Addition of terms to the
controllability matrix with higher powers of the matrix cannot increase the rank of the
matrix because they are linear combination of each other.
Controllability matrix is consisting of four variables; the rank of the matrix should be 4
to be controllable. By using the command ctrb in MATLAB to generate the controllability
matrix. Likewise using rank command we can find the rank. So we will test in simulation
chapter.
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3.4 Pre-compensation
The designed controller meets our transient requirements so, but now we should focus
upon the steady-state error. With respect to the other design methods, where we
feedback the output and compare it to the reference input to compute an error, with a
full-state feedback controller we are feeding back all of the states. We need to compute
what the steady-state value of the states should be, multiply that by the chosen gain,
and use a new value as our "reference" for computing the input. We can do it by adding
a constant gain after the reference input.
Fig 3.3: the cart pendulum system
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Chapter 4
Simulation
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Chapter 4 4.1: Time response
The time response shows how the state of a dynamic system changes with respect to time when
a particular input is applied. Our system consists of differential equations, so we must perform
some integration in order to determine the time response of this dynamic system. For most
systems, especially nonlinear systems or those subject to different input parameters, we have to
carry out integrations numerically. But in case of linear systems MATLAB provides many useful
commands for calculating time responses for many types of inputs.
The time response of a linear dynamic system consists of the sum of the transient response and
steady state response. Transient response depends on the initial conditions while the steady-
state response which depends on the system input. So in the differential equation contains two
terms, one is due to free parameters and other is due to forced parameters.
Frequency Response
Linear time invariant systems are most important because its response linear for state variables.
The property of LTI that the input to the system is sinusoidal, therefore the steady-state output
will also be sinusoidal at the same frequency. They only differs in phase and magnitude. These
magnitude and phase differences as a function of frequency and called as the frequency
response of the system.
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In various ways we can get the frequency response analysis (varying between zero or "DC" to
infinity). We will start with computing the value of the plant transfer function at known
frequencies. If G(s) is the open-loop transfer function of a system and is the desired frequency,
we then plot versus . Then we can use bode plot and Nyquist plot to get rid of complex
frequency values.
Stability
In the pendulum cart system we will use the Bounded Input Bounded Output (BIBO)
definition of stability which states that a system is stable if the output remains bounded in a
linear region for all inputs in the controllable region. Means this system will not diverges for any
input to the systems while working.
To check the system stability the transfer function representation must needed. If all poles of the
transfer function (values of s at which the denominator equals zero) lies in the –VE X-axis then
system is called stable. If any pole has a positive real part, then the system is unstable. If any pair
of poles is on the imaginary axis, then the system is called marginally stable and the system will
oscillate infinite time which is not possible for a practical case. The poles of a LTI system model
can easily be found in MATLAB using the command “POLE”.
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4.2 Controller design:
Take all system
parameters
Choose other
parameters
SYSTEM
ANALYSIS
Gravity
term
Choose the transfer
function
Open loop impulse
response Open loop step
response
Use the feed back
system
Apply PID
CONTROL
Add pre-compensation
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4.3 Simulator Design:
Figure 4.1: simulation of open loop
GET THE POLES FOR
closed LOOP TRANSFER
FUNCTION
DO THE LQR ANALYSIS
SET PENDULUM
COMPONENT
PARAMETER
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Another recursive approach:
Figure 4.2: simulation of closed loop transfer function
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Chapter 5
KALMAN FILTER
(THE ESTIMATOR AND PREDICTOR)
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Chapter 5 4.1 Introduction to kalman filter
Discrete time linear systems basically represented as
5.1
Where;
xj, represent the jth state variables,
a and b are constants
uj represents control input in jth state ;
j is the time variable.
Note that many type of kalman filter don't include the input term (zero initial conditions), and k
is also used to represent time. I have chosen to use j to represent the time variable because we
use the variable k for the Kalman filter gain later in the text. The extension of kalman filter
represented next.
Important properties of kalman filter:
• Data processed through recursive algorithm.
• For a given the set of measurements, it generates optimal estimate of desired outputs.
• Optimal outputs:
– It is the best filter to get the minimal error according to the previous state.
– For non-linear system optimality is very possible using unsent kalman filtering.
• Recursive algorithm:
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– It does not store any previous data, it recursively calculates all every time.
Where,
4.2 Discrete Kalman Filter
By sampling in the time domain the sample at unit time is stored in the state variables and then
the operations will be performed:
• Estimation the state variables nx which is a linear stochastic difference equation
11 kkkk wBuAxx 5.3
– Process noise w will obtained from N(0,Q), where covariance matrix represented
as Q.
• with all measurement parameter which is also real ( mz )
kkk vHxz 5.4
x k F k x k G k u k v k
y k H k x k w k
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1
x k n
u k m
y k p
F k G k H k
v k w k
Q k R k
( )
( )
( )
( ), ( ), ( )
( ), ( )
( ), ( )
is the - dimensional state vector (unknown)
is the - dimensional input vector (known)
is the - dimensional output vector (known, measured)
are appropriately dimensioned system matrices (known)
are zero - mean, white Gaussian noise with (known)
covariance matrices
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– Measurement noise v is obtained from N(0,R), where R is the covariance matrix.
A, Q are n dimension square matrix, B is having n*l configuration and ‘R’ is mxm and ‘H’ is mxn
Time Update (Predictor step):
In the predictor step time will be updated with respect to both control variable and state
variable-
• Expected Update in the value of x in time domain
kkk BuxAx
1ˆˆ 5.5
• Expected update for the covariance matrix:
QAAPP
T
kk 1 5.6
• The simplest method to show the equation 5.5 and 5.6
][)(ˆ)(ˆ 2323 ttutxtx 5.7
][)()( 23
2
2
2
3
2 tttt
Measurement Update (Corrector step)
According to the predicted values we have to update the next time domain quantities
• Value Updated in corrector step:
)ˆ(ˆˆ kkkkk xHzKxx 5.8
– The real error update is kk xHz ˆ
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• Updated error for the covariance matrix
kkk PH)K(IP 5.9
• Compare the equation 5.8 and 5.9
))(ˆ)(()(ˆ)(ˆ 33333
txztKtxtx 5.10
)())(1()( 3
2
33
2 ttKt
The Kalman Gain
• Then Kalman gain Kk for the optimal state:
1)( RHHPHPKT
k
T
kk 5.12
RHHP
HP
T
k
T
k 5.13
• The equation 5.12 can be comparable with simple kalman filter as
2
33
2
3
2
3)(
)()(
t
ttK 5.14
The Jacobian Matrix
To simplify a system equation to use the nonlinear kalman filter we use the Jacobean matrix
• For implementation of a scalar function y=f(x),
xxfy )( 5.15
• For implementation of a vector function y=f(x),
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n
n
nn
n
n x
x
x
f
x
f
x
f
x
f
y
y
1
1
1
1
1
1
)()(
)()(
xx
xx
xJy
5.16
EKF Update Equations
By using the nonlinear kalman filter, the filter operation know as extended kalman filter (EKF)
whose steps are derived above and only the equation given below:
• Predictor step:
),ˆ(ˆ1 kkk f uxx
5.17
QAAPP
T
kk 1 5.18
• Kalman gain:
1)( RHHPHPKT
k
T
kk 5.19
• Corrector step:
))ˆ((ˆˆ kkkkk h xzKxx 5.20
kkk PH)K(IP 5.21
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4.3 EKF ALL STEP AS A GRAPH:
Fig 4.1 Extended kalman filter in graph representation
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Chapter 6
HARDWARE IMPLIMENTATION
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Chapter 6 6.1 The components used to build a working model
1. Arduino mega 2560
2. Inertial measurement unit
3. X-bee
4. Dc motor and shaft encoder.
5. High current Dc motor
6. Lcd
7. Power circuit.
6.2 Arduino mega 2560:
The Arduino Mega 2560 is a arduino based microcontroller board based on the ATmega
microcontroller 2560. It contains 16 analog inputs, 4 UARTs (hardware serial ports), 54 digital
input/output pins (of which 14 can be used as PWM outputs), a 16 MHz crystal oscillator, a
power jack, an ICSP header, a USB connection, as well as a reset button. It contains all on chip
peripherals to support the microcontroller. By connecting it simply to a computer with a USB
cable or with an AC-to-DC adapter or battery (for power purpose), to get started. Most shields
designed for the Arduino Duemilanove or Diecimila is compatible with arduino mega 2560.
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The new feature added in Mega2560 is the FTDI USB-to-serial driver chip which differs
from all preceding boards. At last, it uses the ATmega16U2 (ATmega8U2 is used in revision
1 and revision 2 boards) programmed as a USB-to-serial converter.
The data lines for I2C added SDA and SCL pins that are near to the AREF pin and two other
new pins AREF and ground placed near to the RESET pin, the IOREF that allow the shields
to adapt to the voltage provided from the board. In future, shields will be compatible both
with the board that use the AVR, which operate with 5V and with the Arduino Due that
operate with 3.3V.
Speci f ication:
Microcontroller ATmega2560
Operating Voltage 5V
Input Voltage (recommended) 7-12V
Input Voltage (limits) 6-20V
Digital I/O Pins 54 (of which 15 provide PWM output)
Analog Input Pins 16
DC Current per I/O Pin 40 mA
DC Current for 3.3V Pin 50 mA
Flash Memory 256 KB (8 KB used by bootloader)
SRAM 8 KB
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EEPROM 4 KB
Clock Speed 16 MHz
Table 6.1 :arduino mega 2560 specif icat ion
Memory:
The ATmega2560 is capable to store code in 256 KB of flash memory (of which 8 KB is used
for the bootloader), 8 KB of SRAM and 4 KB of EEPROM (which is used for store variable at run
time).
Input and Output:
All the 54 digital pins on arduino Mega can be used as an digital input or output, using some
functions like digitalWrite(), pinMode(), and digitalRead() functions. They operate at
A input voltage of 5 volts. Maximum current flow in these digital pins is of 40 mA and has an
internal pull-up resistor (disconnected by default) of 20-50 kOhms. Like atmega some pins are
having one or more specialized functions:
Serial: 0 (RX) and 1 (TX); Serial 1: 19 (RX) and 18 (TX); Serial 2: 17 (RX) and 16 (TX); Serial
3: 15 (RX) and 14 (TX). Used to receive (RX) and transmit (TX) TTL serial data. UART 0 is also
connected the ATmega16U2 USB-to-TTL Serial chip used to program the board.
External Interrupts: 2 (interrupt 0), 3 (interrupt 1), 18 (interrupt 5), 19 (interrupt 4), 20
(interrupt 3), and 21 (interrupt 2). These pins can be configured to trigger an interrupt on
a rising or falling edge on a low value, or a change in value. Attach Interrupt() function
gives all details about interrupts
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SPI: 50 (MISO), 51 (MOSI), 52 (SCK), 53 (SS). These pins support SPI communication using
the SPI library.
LED: 13. A led is connected to digital pin 13. When the pin goes HIGH, the LED is on, when
the pin goes LOW, it's off.
PWM: 2 to 13 and 44 to 46. It uses analogWrite() function to Provide 8-bit PWM output.
Pin Diagram:
Fig 6.1: PIN diagram of arduino mega 2560
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Programming
The Arduino Mega can be programmed with the Arduino software which is a open source
software. Arduino is so famoused due to its open and long library. The ATmega2560 on the
Arduino Mega comes pre-burned with a boot loader which helps to upload new code to it
without the using an external hardware programmer. It communicates using the
STK500 protocol which is also used by the atmega.
6.3 Inertial measurement Unit
The IMU is an electronics module consist of more than one module in a single unit, which takes
angular velocity and linear acceleration data as a input and sent to the main processor. The IMU
sensor actually contains three separate sensors. The first one is the accelerometer. To describe
the acceleration about three axes it generates three analog signals and acting on the planes and
vehicle. Because of the physical limitations and thruster system, the significant output sensed
of these accelerations is for gravity. The second sensor is the gyroscope. It also gives three
analog signals. These signals describe the vehicle angular velocities about each of the sensor
axes. It not necessary to place IMU at the vehicle centre of mass, because the angular rate is
not affected by linear or angular accelerations. The data from these sensors is collected by the
microprocessor attached to the IMU sensor through a 12 bit ADC board. The sensor information
communicates via a RS422 serial communications (UART) interface at a rate of about 10 Hz.
The accelerometer and gyroscope within the IMU are mounted such that coordinate
axes of their sensor are not aligned with self-balancing bot. This is the real fact that the two
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sensors in the IMU are mounted in two different orientations according to its orientation of the
axis needed..
Fig 6.2:IMU sensor angle
The accelerometer is manufactured by using a left handed coordinate system. The
transformation algorithm first uses fig 6.2 to align the coordinate axes of the two sensors.
Notice that the gyroscope are now aligned and right handed according to the IMU axis.
Once the accelerometer and gyroscope axes are aligned with the axes of the IMU, then it
should be aligned to vehicle reference frame. Let’s take an example, the unit is mounted on the
wall of the electronics module, and we have to rotate it 45° with respect to the horizontal. So
according to the Euler’s axis transformation, the angle it deviates
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from the vehicle axes. This was calculated by taking the assumption that over the 30 inches
from the back to the front of the of the IMU sensor the reference move inward 4 inches.
Using information of angel transformation, along with the orientation with which the IMU is
mounted on the vehicle of the euler angle transformation allows to form a direction cosine
matrix that is used to convert the IMU coordinate measurement frame to the vehicle
coordinate frame. Figure 6.3 illustrates the orientation with which the IMU is mounted with
the three sensors.
Fig 6.3: Euler angle transformtation.
Transformed IMU coordinate axes compared to vehicle coordinate axes
Following is the order of transformations:
1) Rotating alpha + 90° about the x-axis to align the IMU z-axis with the Vehicle frame(Z-
axis).
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6.1
2) Rotate beta + 90° about the z-axis to align IMU and vehicle coordinate frames.
6.2
3) So the complete transformation from IMU to vehicle coordinates is given by
6.3
4) Inserting numerical values for the angles ([alpha] = 45°, [beta] = 7.5946°), produces:
6.4
In order to transform the sensor data from the individual sensor frames into the vehicle
frame, first use the simple sign/axis transformations shown in (6.1), and then pre-multiply
the modified sensor data by the transformation matrix given in (6.3). It would also be
possible to combine the initial transformations from (6.1) into the transformation in (6.3),
but the initial axes of the two sensors in the IMU are different on their own axes frame,
there would be a different transformation matrix for each sensor. To simplify the derivation,
the program first aligns the sensor axes using (6.1), and then uses the second
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transformation matrix from (2.6) to transform the data of the accelerometer and angular
rate sensors from IMU to vehicle coordinates.
6.4 Dc motor and Shaft encoder
Figure 6.4 High torque motor
Here I am using the 85RPM 37DL Gear Motor with shaft Encoder, which is a high performance DC
gear motor and contains magnetic Hall Effect quadrature shaft encoder. Shaft Encoder gives 90
pulses per revolution of the output shaft per channel. Motor can run from 4V to 12V supply. For shaft
encoder signals, Sturdy 2510 relimate connector is used and Shaft encoders are used in applications
that requires motor’s angle of rotation such as robotics, CNC etc.
Specifications:
o Supply: 12V (Motor runs smoothly from 4V to12V)
o Shaft encoder resolution: 90 quadrature pulses per revolution of the output shaft
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o RPM: 85RPM @ 12V; 42RPM @6V
o Stall Current: 1.8A @ 12V; 0.8A @ 6V
o Stall Torque: 21kg/cm @ 12V; 10kg/cm @ 6V
o Shaft encoder supply: 3.3V or 5V
o Shaft encoder current requirement: 5 to 10mA
o Shaft diameter: 6mm
o Gear ratio: 1:30
o Table 6.2:motor specification
Interfacing
We are using Sturdy 2510 relimate to give power to motor and take the output signal of shaft
encoder. Shaft encoder needs a supply voltage of 3.3V or 5V DC. 1K to 2.2K pull-up resistor is
connected between Vcc and Channel A and Channel B.
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table 6.3:interfacing of motor
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Chapter 7
Real Time implementation
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Chapter 7
7.1 Algorithm to control SBB vertically upward
Take the necessary header files and global variables to support the main
software.
Configure PID angle and speed of the motor as well as PID constants.
Take the read values of the IMU sensor and store it in a matrix up to 100
samples
Configure the start button, stop button and device configure button.
Configure PID module to update it.
Set digital pins, serial pins and PWM pins on the arduino board
Set the kalman filter and update its weight matrix and coefficient.
Initialize PID speed, PID angle and Time action module.
Update the IMU sensor and update the IMU matrix.
Then we will face three conditions
1) When debug->
a) calibrate the previous values and update the previous
declared variables.
2) When started->
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A. calculate the PID angle and motor speed.
B. According to the angel check motor speed regularly and
update the IMU sensor.
C. If steering is not applied then set the external offset to
zero
D. Else refer to SBB console program.
3) When stoped-> do the power off. Fall automatically.
7.2 Algorithm to steer the SBB by remote control
Take the necessary header files and global variables to support the main
software.
Configure PID angle and speed of the motor as well as PID constants.
Take the read values of the IMU sensor and store it in a matrix up to 100
samples
Configure the start button, stop button and device configure button.
Configure PID module to update it.
Set digital pins, serial pins and PWM pins on the arduino board
Set the kalman filter and update its weight matrix and coefficient.
Set the remote of the buttons as well as the UART 1.
If any button is pressed make its coefficient to 100.
Add it to the next time action.
Make the PID mode aggressive.
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Display it on the LCD
Check the control parameter according to the change in PID angle.
7.3 Other files used in the implementation
I. Remote_communication(using serial communication)
II. Eeprom.h (to store the update values)
III. Serial_communication
IV. Motor.h
V. Shaft_encoder.h
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Chapter 8
Result and Graph
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Chapter 8
8.1 result for controller design
According to the system model the open loop response of the system is not controllable
and the graph is giving peak means the angle and the amplitude goes on increasing which is not
BIBO stable. Fig 8.1 describes the same.
Fig 8.1 open loop impulse response and open loop step response
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To find the root locus of the open loop transfer function we used the specific
MATLAB command. The graph shows that the pole are on the +VE half of the X-
axis which leads to system is unstable.
Fig 8.2 pole location and root locus
Design of a closed loop transfer function using the state space model can be done
in two ways. By using LQR algorithm and also by placing the closed loop pole
location on the desired position. The next two figures describe the same. Figure
8.3 describes the LQR model while in figure 8.4 shows LQR model with PID tuning.
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Fig 8.3 Control using LQR model
Fig 8.4 Control using LQR model with tuning parameters
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As we discussed in chapter two though it is a feedback system, two take the error
with respect to same reference we have to add a pre-compensation before the
control input. The effect of pre-compensation is described in the following graph
Fig 8.5 control in addition of pre-compensation
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8.2 simulation results
By doing the simulation using MATLAB we get into the conclusion that the system function and
parameter we derived as a reference for self-balancing bot work well. So in the next step we
can proceed through the design. With open loop transfer function the animated system is
unstable as shown in fig 8.6 whereas for a close loop transfer function the system is stable.
Fig 8.6 cart pendulum system in unstable condition
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Fig 8.7 self-balanced Pendulum cart system
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8.3. Hardware simulation result.
The following figure is taken from XCTU which is an interface to communicate
with computer from the arduino board with the help of serial communication. It
shows the output of the IMU sensor in form of yaw roll and pitch.
Fig 8.8 output of IMU sensor
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Shaft encoder calculates the rotation of the wheel according to the values obtain from channel
and channel b.both of them with a 90 degree phase shift each giving 90 count per 1 revolution
making total of 180 counts
Fig 8.9 output of shaft encoder
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After all the derivations and simulations the final working model is derived which
consist of all the components. fig 8.10 shows the working model on the upward
direction.
Fig 8.10 working model of SBB
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Chapter 9
CONCLUSION
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Chapter 9
Conclusion: To build a self-balancing robot we first derived the system equation then check its real
time response (both time and frequency). Then we designed a PID controller to control the
close loop function. We checked the controllability and set the pole location. Then we used
kalman filter as estimator and predictor. Then by choosing the appropriate components we
analyse their simulation sucessfully
The above test steps are successful, then we are near to build a SBB. The easiest way to
tune a PID controller is to tune the P, I and D parameters one at a time. It was done successfully.
The stability of the SBB may be improved if you use a properly designed gearbox that is having
negligible gear backlash.
So by implementation all of these concepts and avoid the errors that we came across the
self-balancing bot is completely build. We can make Segway and ball bot as a application of
self-balancing bot
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Bibliography
[1]http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=SimulinkMod
eling
[2]http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=SystemModeli
ng
[3] http://playground.arduino.cc/Main/RotaryEncoders
[4] http://blog.wolfram.com/2011/01/19/stabilized-inverted-pendulum/
[5] Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems”,
Transaction of the ASME--Journal of Basic Engineering, pp. 35-45 (March 1960).
[6] Welch, G and Bishop, G. 2001. “An introduction to the Kalman Filter”,
http://www.cs.unc.edu/~welch/kalman/
[7] Maybeck, P. S. 1979. “Stochastic Models, Estimation, and Control, Volume 1”, Academic
Press, Inc.
[8] http://www.kerrywong.com/2012/03/08/a-self-balancing-robot-i
[9] http://diydrones.com/profiles/blog/show?id=705844%3ABlogPost%3A23188
[10] Lei Guo , Hong Wang , “PID controller design for output PDFs of stochastic systems using
linear matrix inequalities”, IEEE Transactions on Systems, Man, and Cybernetics, Part B
01/2005; 35:65-71. pp.65-71.
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